4.7 Observables from Localized Sources
75
δ θ D = −κ∂ ζ D + ˆ
n
i ˆ
θ
0i D + O(θ
2
) ,
(4.7.56)
δ θ s ◦ = −κ∂ ζ s ◦ + O(θ
2
) ,
(4.7.57)
δ θ s = −κ∂ ζ s − 2iαs + O(θ
2
) ,
(4.7.58)
δ θ θ = −κ∂ ζ θ + O(θ
2
) ,
(4.7.59)
δ θ σ = −κ∂ ζ σ − 2iασ + O(θ
2
) .
(4.7.60)
Note that these transformations abide to the criteria of cosmological observables,
i.e. they solely depend on the LLTs at the observer and source positions. In fact,
in the present case there is actually no dependence on the LLT parameter at the
source position, implying in particular that these quantities do not depend on the
source’s 4-velocity. In contrast, this is not the case of the luminosity distance, given
by Etherington’s distance-duality equation
D L := (1 + z)
2 D ≡ e
2ζ D ,
(4.7.61)
because the redshift is sensitive to LLTs at the source.
4.7.2 Volume and Source Number Density
We now wish to find the relation between the physical volume occupied by some
source dV (γ(ζ, ϑ)) in its rest-frame e 0 (γ(ζ, ϑ)) and the corresponding observed solid
angle d(ϑ) and observed redshift interval dz(ζ, ϑ). This information is needed
in order to infer a number density from number counts of localized sources, e.g.
galaxies. We first split dV into an area element d A that is normal to n
i
(ζ, ϑ) and a
length element dL that goes along that direction
dV := d A dL .
(4.7.62)
From the previous subsection we already have the relation between the are and the
solid angle d A = D
2 d, so we look for the dL ∼ dz relation. The length element
dL can be defined implicitly through the corresponding derivative operator acting
on space-time fields X evaluated on the geodesic
∂ L X := n
i
∂ i X ,
(4.7.63)
which is (minus) the spatial part of ∂ := ∂ 0 − n
i
∂ i . To relate ∂ L to ∂ z , we need to go
back to a generic λ-parametrization of the line manifold L and note that Eq. (4.2.1)
gives
∂ = (ω)
−1
∂ λ γ
μ
∂ μ ,
(4.7.64)
so for fields on L this reduces to
75
δ θ D = −κ∂ ζ D + ˆ
n
i ˆ
θ
0i D + O(θ
2
) ,
(4.7.56)
δ θ s ◦ = −κ∂ ζ s ◦ + O(θ
2
) ,
(4.7.57)
δ θ s = −κ∂ ζ s − 2iαs + O(θ
2
) ,
(4.7.58)
δ θ θ = −κ∂ ζ θ + O(θ
2
) ,
(4.7.59)
δ θ σ = −κ∂ ζ σ − 2iασ + O(θ
2
) .
(4.7.60)
Note that these transformations abide to the criteria of cosmological observables,
i.e. they solely depend on the LLTs at the observer and source positions. In fact,
in the present case there is actually no dependence on the LLT parameter at the
source position, implying in particular that these quantities do not depend on the
source’s 4-velocity. In contrast, this is not the case of the luminosity distance, given
by Etherington’s distance-duality equation
D L := (1 + z)
2 D ≡ e
2ζ D ,
(4.7.61)
because the redshift is sensitive to LLTs at the source.
4.7.2 Volume and Source Number Density
We now wish to find the relation between the physical volume occupied by some
source dV (γ(ζ, ϑ)) in its rest-frame e 0 (γ(ζ, ϑ)) and the corresponding observed solid
angle d(ϑ) and observed redshift interval dz(ζ, ϑ). This information is needed
in order to infer a number density from number counts of localized sources, e.g.
galaxies. We first split dV into an area element d A that is normal to n
i
(ζ, ϑ) and a
length element dL that goes along that direction
dV := d A dL .
(4.7.62)
From the previous subsection we already have the relation between the are and the
solid angle d A = D
2 d, so we look for the dL ∼ dz relation. The length element
dL can be defined implicitly through the corresponding derivative operator acting
on space-time fields X evaluated on the geodesic
∂ L X := n
i
∂ i X ,
(4.7.63)
which is (minus) the spatial part of ∂ := ∂ 0 − n
i
∂ i . To relate ∂ L to ∂ z , we need to go
back to a generic λ-parametrization of the line manifold L and note that Eq. (4.2.1)
gives
∂ = (ω)
−1
∂ λ γ
μ
∂ μ ,
(4.7.64)
so for fields on L this reduces to